Applications of Hamming Distance to the Analysis of Block Designs

Abstract

Hamming distance is used in coding theory for binary strings and can be used to define a rank correlation. Using Hamming distance between rankings, we generate rank-based tests for the analysis of general complete and incomplete block designs. The methodology used here relies on using the concept of compatibility to extend the definition of Hamming distance between complete rankings to those which may have missing observations or ties. In the incomplete case, the asymptotic distribution of the resulting statistic relies on the eigen values of two matrices; one the so-called information matrix of a block design, and the other a matrix whose eigen values are determined through the use of orthogonal polynomials. In one section, there is a recall on the notion of compatibility and use it to extend the Hamming distance between two complete rankings to the situations where either missing observations or ties may be present. The resulting tests have forms which are easily calculated with asymptotic distributions that are linear combinations of independent chi squares. In the case of incomplete blocks, the coefficients of these linear combinations are products of the eigen values of two (t x t) matrices. When the number of missing observation is the same for each block, one set of these eigen values is completely specified, while the eigen values of the other matrix are well known for many designs. Once the coefficients are determined, the critical values can be approximated.